On Finsler manifolds with the Chern–Rund connection whose projectivised tangent bundle has the Sasaki type metric

It is shown that if the projectivised tangent bundle PTM of an mm-dimensional Finsler manifold MM with the Chern–Rund connection has the Sasaki type metric, PTM is locally isometric Em×Sm−1(4)Em×Sm−1(4) for m>2m>2 and flat for m=2m=2, and moreover MM is a Riemannian manifold.